Decay of Isotropic Turbulence in the Direct-Interaction Approximation

1964 ◽  
Vol 7 (7) ◽  
pp. 1030 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Isotropic Turbulence ◽  
Direct Interaction ◽  
Direct Interaction Approximation ◽  
Interaction Approximation

Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory

1989 ◽  
Vol 208 ◽  
pp. 91-114 ◽  
Author(s):  
W. D. Mccomb ◽  
V. Shanmugasundaram ◽  
P. Hutchinson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Direct Interaction ◽  
Reynolds Numbers ◽  
Time Correlation ◽  
General Belief ◽  
Direct Interaction Approximation ◽  
The Difference ◽  
Decaying Turbulence ◽  
Interaction Approximation

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

Self-consistent effective-medium parameters for oceanic internal waves

1984 ◽  
Vol 146 ◽  
pp. 253-270 ◽  
Author(s):  
R. J. Dewitt ◽  
Jon Wright
Keyword(s):  
Internal Waves ◽  
Direct Interaction ◽  
Decay Rates ◽  
The Self ◽  
Time Behaviour ◽  
Equilibrium Time ◽  
Self Consistent ◽  
Direct Interaction Approximation ◽  
Interaction Approximation ◽  
Extra Parameter

In this paper we apply a formalism introduced in a previous paper to write down a self-consistent set of equations for the functions that describe the near-equilibrium time behaviour of random oceanic internal waves. These equations are based on the direct-interaction approximation. The self-consistent equations are solved numerically (using the Garrett-Munk spectrum as input) and the results are compared to parameters obtained in the weak-interaction approximation (WIA). The formalism points out that an extra parameter that is implicitly vanishingly small in the WIA has a significant effect on decay rates when computed self-consistently. We end by mentioning possible future self-consistent calculations that would improve upon our own.

KOLMOGOROV CONSTANT IN LARGE SPACE DIMENSIONS

2002 ◽  
Vol 16 (32) ◽  
pp. 4839-4845 ◽  
Author(s):  
MALAY K. NANDY
Keyword(s):  
Eddy Viscosity ◽  
Energy Equation ◽  
Space Dimension ◽  
Direct Interaction ◽  
Large Space ◽  
Distant Interaction ◽  
Direct Interaction Approximation ◽  
Dimension Expansion ◽  
Kolmogorov Constant ◽  
Interaction Approximation

A large d (space dimension) expansion together with the ∊-expansion is implemented to calculate the Kolmogorov constant from the energy equation of Kraichnan's direct-interaction approximation using the Heisenberg's eddy-viscosity approximation and Kraichnan's distant-interaction algorithm. The Kolmogorov constant C is found to be C = C0 d1/3 in the leading order of a 1/d expansion. This is consistent with Fournier, Frisch, and Rose. The constant C0 evaluated in the above scheme, is found to be C0 = (16/27)1/3.

Lagrangian velocity covariance in helical turbulence

1977 ◽  
Vol 81 (2) ◽  
pp. 385-398 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Correlation Time ◽  
Perturbation Expansion ◽  
Direct Interaction ◽  
Helical Turbulence ◽  
Quantitative Accuracy ◽  
Lagrangian Velocity ◽  
Thin Spherical Shell ◽  
Direct Interaction Approximation ◽  
Interaction Approximation ◽  
Substantial Factor

The effect of helicity on the Lagrangian velocity covarianceUL(t) in isotropic, normally distributed turbulence is examined by computer simulation and by a renormalized perturbation expansion forUL(t). The first term of the latter represents Corrsin's (1959) conjecture (extrapolated to allt), which relatesUL(t) to the Eulerian covariance and the distributionG(x, t) of fluid-element displacement. Truncation of the expansion at the first term yields the direct-interaction approximation forG(x, t). The expansion suggests that with or without helicity Corrsin's conjecture is valid ast→ ∞ and that in either caseUL(t) behaves asymptotically like$t^{-(r+\frac{3}{2})}$if the spectrum of the Eulerian field varies likekr+2at small wavenumbers. Corrsin's conjecture breaks down at small and moderatetif there is strong helicity while remaining accurate at alltin the mirror-symmetric case. Computer simulations for a frozen Eulerian field with spectrum confined to a thin spherical shell inkspace indicate that strong helicity induces an increase in the Lagrangian correlation time by a factor of approximately three. Direct-interaction equations are constructed for the Lagrangian space-time covariance and the resulting prediction forUL(t) is compared with the simulations. The effect of helicity is well represented quantitatively by the direct-interaction equations for small and moderatetbut not for larget. These frozen-field results imply good quantitative accuracy at alltin time-varying turbulence whose Eulerian correlation time is of the order of the eddy-circulation time. In turbulence with weak helicity, the directinteraction equations imply that the Lagrangian correlation of vorticity with initial velocity is more persistent thanUL(t), by a substantial factor.

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